From a class of 14 boys and 10 girls, 10 students are to be chosen for a competition, at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?
2 girls who won the prize last year are surely to be taken. So, we have to make a selection of 8 students out of 14 boys and 8 girls, choosing at least 4 boys and at least 2 girls.
Thus, we may choose:
(4 boys, 4 girls) or (5 boys, 3 girls) or (6 boys, 2 girls)
Therefore, the required number of ways = (14C4 × 8C4) + (14C5 × 8C3) + (14C6 × 8C2)